3/21/2015 Assignment Previewer Practice Exam # 2 (2.7­3.9) (6968235) Due: Thu Mar 26 2015 11:59 PM PDT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Question 1. Question Details SCalcET7 2.8.027. [1735792] ­ Find the derivative of the function using the definition of derivative. g(x) = 1 − x g'(x) = State the domain of the function. (Enter your answer using interval notation.) State the domain of its derivative. (Enter your answer using interval notation.) Solution or Explanation g'(x) = lim h → 0 g(x + h) − g(x) h 1 − (x + h) − = lim h → 0 1 − (x + h) + 1 − x 1 − (x + h) + 1 − x [1 − (x + h)] − (1 − x) = lim h → 0 h 1 − (x + h) + 1 − x −h = lim h → 0 h 1 − (x + h) + 1 − x −1 = lim 1 − (x + h) + h → 0 = 1 − x h 1 − x −1 2 1 − x Domain of g = (−∞, 1], domain of g' = (−∞, 1). http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 1/12 3/21/2015 2. Assignment Previewer Question Details SCalcET7 2.8.037. [1637474] ­ SCalcET7 3.1.069. [1783058] ­ The graph of f is given. State the numbers at which f is not differentiable. x= ­4 (smaller value) x= 0 (larger value) Solution or Explanation Click to View Solution 3. Question Details Consider the following function. f(x) = |x2 − 9| (a) Find a formula for f '. if |x| > 3 if |x| < 3 f '(x) = For what values of x is the function not differentiable? (Enter your answers as a comma­separated list.) x = (b) Sketch the graph of f. http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 2/12 3/21/2015 Assignment Previewer Sketch the graph of f ' http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 3/12 3/21/2015 Assignment Previewer Solution or Explanation (a) Note that x2 − 9 < 0 for x2 < 9 ⇔ |x| < 3 ⇔ −3 < x < 3. So x2 − 9 if x ≤ −3 f(x) = −x2 + 9 if −3 < x < 3 x2 − 9 if x ≥ 3 2x if x < −3 2x if |x| > 3 f '(x) = −2x if −3 < x < 3 = −2x if |x| < 3 2x if x > 3 To show that f '(3) does not exist we investigate lim f(3 + h) − f(3) h h → 0 [−(3 + h)2 + 9] − 0 by computing the left­ and right­hand derivatives. f(3 + h) − f(3) lim and = lim = h → 0− (−6 − h) = −6 h h h → 0− 2 2 f(3 + h) − f(3) lim [(3 + h) − 9] − 0 = lim 6h + h = lim (6 + h) = 6. f ' +(3) = lim = h + + h h h → 0+ h → 0 h → 0 h → 0+ f(3 + h) − f(3) Since the left and right limits are different, lim does not exist, that is, f '(3) does not exist. Similarly, h h → 0 f ' −(3) = lim h → 0− f '(−3) does not exist. Therefore, f is not differentiable at 3 or at −3. (b) http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 4/12 3/21/2015 4. Assignment Previewer Question Details SCalcET7 3.2.502.XP. [1836132] ­ SCalcET7 3.2.020. [1835633] ­ SCalcET7 3.3.015. [1835932] ­ Find the derivative of the function below in two ways. F(x) = x − 3x x x (a) by using the Quotient Rule F'(x) = (b) by simplifying first F'(x) = (c) Which method appears to be simpler for this problem? the Quotient Rule simplifying first Solution or Explanation Click to View Solution 5. Question Details Differentiate. (Assume c is a constant.) z = u3/2(u + ceu) z' = Solution or Explanation Click to View Solution 6. Question Details Differentiate. f(x) = 2xex csc x f '(x) = Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 5/12 3/21/2015 7. Assignment Previewer Question Details SCalcET7 3.3.022. [1836395] ­ SCalcET7 3.4.023. [1799833] ­ Find an equation of the tangent line to the curve at the given point. y = 3ex cos x, (0, 3) y = Solution or Explanation Click to View Solution 8. Question Details Find the derivative of the function. y = 5 + 8e9x y' = Solution or Explanation y = 5 + 8e9x y' = 1 d (5 + 8e9x )−1/2 (5 + 8e9x ) = 2 2 dx 1 (8e9x · 9) = 5 + 8e9x 36e9x 5 + 8e9x 9. Question Details SCalcET7 3.4.031. [1835882] ­ SCalcET7 3.4.033. [1835657] ­ Find the derivative of the function. y = sin(tan 6x) y' = Solution or Explanation Click to View Solution 10. Question Details Find the derivative of the function. y = 7sin πx y' = Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 6/12 3/21/2015 11. Assignment Previewer Question Details SCalcET7 3.4.041. [1836073] ­ SCalcET7 3.4.044. [1836022] ­ SCalcET7 3.5.008.MI. [1645360] ­ SCalcET7 3.5.019. [1836234] ­ Find the derivative of the function. 2 f(t) = cos2(ecos t) f '(t) = Solution or Explanation Click to View Solution 12. Question Details Find the derivative of the function. y' = Solution or Explanation Click to View Solution 13. Question Details Find dy/dx by implicit differentiation. 6x3 + x2y − xy3 = 3 y' = Solution or Explanation Click to View Solution 14. Question Details Find dy/dx by implicit differentiation. ey cos x = 9 + sin(xy) y' = Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 7/12 3/21/2015 15. Assignment Previewer Question Details SCalcET7 3.6.004. [1836239] ­ SCalcET7 3.6.007. [1799716] ­ SCalcET7 3.6.011. [1835580] ­ Differentiate the function. f(x) = ln(36 sin2x) f '(x) = Solution or Explanation Click to View Solution 16. Question Details Differentiate the function. f(x) = log10(x9 + 7) f '(x) = Solution or Explanation f(x) = log10(x9 + 7) f '(x) = 1 d 9x8 (x9 + 7) = (x9 + 7)ln 10 dx (x9 + 7)ln 10 17. Question Details Differentiate the function. g(x) = ln x x2 − 5 g'(x) = Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 8/12 3/21/2015 18. Assignment Previewer Question Details SCalcET7 3.6.041. [1799740] ­ Use logarithmic differentiation to find the derivative of the function. y = x − 4 x8 + 5 y' = Solution or Explanation y = x − 4 x8 + 5 y' = y ln y = ln 1 4x7 − 8 2(x − 4) x + 5 x − 4 x8 + 5 1/2 ln y = y' = x − 4 x8 + 5 1 1 ln(x − 4) − ln(x8 + 5) 2 2 1 1 1 1 1 y' = − 8 · 8x7 y 2 x − 4 2 x + 5 1 4x7 − 8 2x − 8 x + 5 19. Question Details SCalcET7 3.6.044.MI. [1836262] ­ Use logarithmic differentiation to find the derivative of the function. y = x4 cos x y' = Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 9/12 3/21/2015 20. Assignment Previewer Question Details SCalcET7 3.7.007. [1815468] ­ The height (in meters) of a projectile shot vertically upward from a point 4 m above ground level with an initial velocity of 21.5 m/s is h = 4 + 21.5t − 4.9t2 after t seconds. (Round your answers to two decimal places.) (a) Find the velocity after 2 s and after 4 s. v(2) = 1.9 m/s v(4) = ­17.7 m/s (b) When does the projectile reach its maximum height? 2.19 s (c) What is the maximum height? 27.58 m (d) When does it hit the ground? 4.57 s (e) With what velocity does it hit the ground? ­23.25 m/s Solution or Explanation (a) h(t) = 4 + 21.5t − 4.9t2 v(t) = h'(t) = 21.5 − 9.8t. the velocity after 2 s is v(2) = 21.5 − 9.8(2) = 1.9 m/s and after 4 s is v(4) = 21.5 − 9.8(4) = −17.7 m/s. (b) The projectile reaches its maximum height when the velocity is zero. v(t) = 0 ⇔ 21.5 − 9.8t = 0 ⇔ 21.5 t = = 2.19 s. 9.8 (c) The maximum height occurs when t = 2.19. h(2.19) = 4 + 21.5(2.19) − 4.9(2.19)2 = 27.58 m. (d) The projectile hits the ground when h = 0 ⇔ 4 + 21.5t − 4.9t2 = 0 ⇔ t = −21.5 ± 21.52 − 4(−4.9)(4) 2(−4.9) t = tf ≈ 4.57 s [since t ≥ 0]. (e) The projectile hits the ground when t = tf. Its velocity is v(tf) = 21.5 − 9.8tf ≈ −23.25 m/s [downward]. 21. Question Details SCalcET7 3.9.005.MI. [1646152] ­ A cylindrical tank with radius 7 m is being filled with water at a rate of 4 m3/min. How fast is the height of the water increasing? m/min Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 10/12 3/21/2015 22. Assignment Previewer Question Details SCalcET7 3.9.014. [2727122] ­ At noon, ship A is 170 km west of ship B. Ship A is sailing east at 40 km/h and ship B is sailing north at 15 km/h. How fast is the distance between the ships changing at 4:00 PM? km/h Solution or Explanation Given: at noon, ship A is 170 km west of ship B; ship A is sailing east at 40 km/hr and ship B is sailing north at 15 km/h. If we let t be time (in hours), x be the distance traveled by ship A (in km), and y be the distance traveled by ship B (in km), then we are given that dx/dt = 40 km/h and dy/dt = 15 km/h. Unknown: the rate at which the distance between the ships is changing at 4:00 PM. If we let x be the distance between the ships, then we want to find dz/dt when t = 4h. z2 = (170 − x)2 + y2 2z dz dx dy = 2(170 − x) − + 2y dt dt dt At 4:00 PM, x = 4(40) = 160 and y = 4(15) = 60 So z = (170 − 160)2 + 602 = 10 37 . (−10)(40) + 60(15) 50 dz 1 dx dy = (x − 170) + y = = ≈ 8.2199 km/h. 10 37 dt z dt dt 37 23. Question Details SCalcET7 3.9.017. [1646109] ­ A man starts walking north at 2 ft/s from a point P. Five minutes later a woman starts walking south at 3 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking? (Round your answer to two decimal places.) 4.98 ft/s Solution or Explanation Click to View Solution http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 11/12 3/21/2015 24. Assignment Previewer Question Details SCalcET7 3.9.028. [1647667] ­ A kite 100 ft above the ground moves horizontally at a speed of 4 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out? 1/100 rad/s Solution or Explanation Click to View Solution Assignment Details Name (AID): Practice Exam # 2 (2.7­3.9) (6968235) Feedback Settings Submissions Allowed: 100 Before due date Category: Homework Question Score Code: Assignment Score Locked: Yes Publish Essay Scores Author: Mkrtchyan, Tigran ( mkrtcht@lamission.edu ) Question Part Score Last Saved: Mar 21, 2015 03:01 AM PDT Mark Permission: Protected Add Practice Button Randomization: Person Help/Hints Which graded: Last Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response http://www.webassign.net/v4cgimkrtcht@lamission/assignments/preview.tpl?d=20150321100314mkrtcht@lamission642034214 12/12